Methods and apparatus for helical reconstruction for multislice CT scan

ABSTRACT

One embodiment of the present invention is a method for imaging an object with a computed tomographic (CT) imaging system that includes steps of helically scanning the object with a multi-slice CT imaging system to acquire attenuation measurements of the object, the measurements including more than two conjugate samples for estimation of a projection at a plane of reconstruction of the object; and filtering and backprojecting the attenuation measurements of the object, including the more than two conjugate samples, to reconstruct at least one image slice of the object. An improved sampling pattern and better use of the attenuation samples obtained during a scan is thus provided.

BACKGROUND OF THE INVENTION

This invention relates generally to methods and apparatus for computedtomographic (CT) imaging, and more specifically to methods and apparatusfor acquiring and reconstructing helically scanned, medical CT imagesusing a multi-slice CT imaging system.

In at least one known computed tomography (CT) imaging systemconfiguration, an x-ray source projects a fan-shaped beam which iscollimated to lie within an X-Y plane of a Cartesian coordinate systemand generally referred to as the “imaging plane”. The x-ray beam passesthrough the object being imaged, such as a patient. The beam, afterbeing attenuated by the object, impinges upon an array of radiationdetectors. The intensity of the attenuated beam radiation received atthe detector array is dependent upon the attenuation of the x-ray beamby the object. Each detector element of the array produces a separateelectrical signal that is a measurement of the beam attenuation at thedetector location. The attenuation measurements from all the detectorsare acquired separately to produce a transmission profile.

In known third generation CT systems, the x-ray source and the detectorarray are rotated with a gantry within the imaging plane and around theobject to be imaged so that the angle at which the x-ray beam intersectsthe object constantly changes. A group of x-ray attenuationmeasurements, i.e., projection data, from the detector array at onegantry angle is referred to as a “view”. A “scan” of the objectcomprises a set of views made at different gantry angles, or viewangles, during one revolution of the x-ray source and detector. In anaxial scan, the projection data is processed to construct an image thatcorresponds to a two dimensional slice taken through the object. In ahelical scan, a table on which the object is resting moves so that theobject itself moves though the imaging plane while it is being scanned.A multi-slice CT imaging system has a plurality of parallel detectorrows configured to acquire attenuation measurements corresponding to oneor more two-dimensional image slices of an object. The number of imageslices and the thicknesses represented by the slices is dependent uponhow (and whether) attenuation measurements from the parallel detectorrows are combined.

One method for reconstructing an image from a set of projection data isreferred to in the art as the filtered back projection technique. Thisprocess converts the attenuation measurements from a scan into integerscalled “CT numbers” or “Hounsfield units”, which are used to control thebrightness of a corresponding pixel on a cathode ray tube display.

Helical reconstruction algorithms for multi-slice CT have been a focusfor many studies. In one known CT imaging system, a reconstructionalgorithm is implemented for two special helical pitches: 3:1 and 6:1.This algorithm utilizes two conjugate samples from different detectorrows to estimate projection samples at a reconstruction plane usinglinear interpolation. Although this method performs satisfactorily inmany cases, it has a number of shortcomings. First, the sampling patternis not always optimum because only two samples on either side of theplane of reconstruction are selected. For example, samples that arecloser to the reconstruction plane but located on the same side of theplane will not be utilized. Second, a 3:1 helical pitch is non-optimalfor projection sampling, because the first and last detector rowsmeasure identical ray paths, reducing the amount of non-redundantinformation that is acquired. In fact, in one known helicalreconstruction implementation, measured projections (after calibration)of these two rows are summed first before reconstruction takes place.Third, sharp structures in the original object (along a z-axis) aresuppressed and degraded slice sensitivity profiles are obtained becauselinear interpolation suppresses high frequency information in thesampled data.

It would therefore be desirable to provide methods and apparatus forhelical reconstruction in multi-slice CT imaging systems that overcomethe above-described shortcomings of known image reconstruction systems.

BRIEF SUMMARY OF THE INVENTION

There is therefore provided, in one embodiment of the present invention,a method for imaging an object with a computed tomographic (CT) imagingsystem that includes steps of helically scanning the object with amulti-slice CT imaging system to acquire attenuation measurements of theobject, the measurements including more than two conjugate samples forestimation of a projection at a plane of reconstruction of the object;and filtering and backprojecting the attenuation measurements of theobject, including the more than two conjugate samples, to reconstruct atleast one image slice of the object.

This embodiment and others provide, among other advantages, an improvedsampling pattern and better use of the attenuation samples obtainedduring a scan.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a pictorial view of a CT imaging system.

FIG. 2 is a block schematic diagram of the system illustrated in FIG. 1.

FIG. 3 is an illustration of a sampling pattern for image reconstructionin an embodiment of the invention employing a 2.5:1 helical pitch scan.

FIG. 4 is an illustration of interpolation, extrapolation, and weightedinterpolation-extrapolation techniques to estimate an attenuation valueat a point x.

FIG. 5 is a graph of slice sensitivity profile measurements using a thinslice phantom, comparing a slice profile using a known techniqueutilizing a 3:1 helical pitch scan to an embodiment of the presentinvention utilizing a 2.5:1 helical pitch scan.

FIG. 6 is a drawing of a curved plane of reconstruction.

DETAILED DESCRIPTION OF THE INVENTION

Referring to FIGS. 1 and 2, a computed tomograph (CT) imaging system 10is shown as including a gantry 12 representative of a “third generation”CT scanner. Gantry 12 has an x-ray source 14 that projects a beam ofx-rays 16 toward a detector array 18 on the opposite side of gantry 12.Detector array 18 is formed by detector elements 20 which together sensethe projected x-rays that pass through an object 22, for example amedical patient. Detector array 18 may be fabricated in a single sliceor multi-slice configuration. Each detector element 20 produces anelectrical signal that represents the intensity of an impinging x-raybeam and hence the attenuation of the beam as it passes through patient22. During a scan to acquire x-ray projection data, gantry 12 and thecomponents mounted thereon rotate about a center of rotation 24.

Rotation of gantry 12 and the operation of x-ray source 14 are governedby a control mechanism 26 of CT system 10. Control mechanism 26 includesan x-ray controller 28 that provides power and timing signals to x-raysource 14 and a gantry motor controller 30 that controls the rotationalspeed and position of gantry 12. A data acquisition system (DAS) 32 incontrol mechanism 26 samples analog data from detector elements 20 andconverts the data to digital signals for subsequent processing. An imagereconstructor 34 receives sampled and digitized x-ray data from DAS 32and performs high speed image reconstruction. The reconstructed image isapplied as an input to a computer 36 which stores the image in a massstorage device 38.

Computer 36 also receives commands and scanning parameters from anoperator via console 40 that has a keyboard. An associated cathode raytube display 42 allows the operator to observe the reconstructed imageand other data from computer 36. The operator supplied commands andparameters are used by computer 36 to provide control signals andinformation to DAS 32, x-ray controller 28 and gantry motor controller30. In addition, computer 36 operates a table motor controller 44 whichcontrols a motorized table 46 to position patient 22 in gantry 12.Particularly, table 46 moves portions of patient 22 through gantryopening 48. During a helical scan, this motion occurs while scanning istaking place.

In one embodiment of the present invention, various shortcomings ofknown CT image reconstruction systems are overcome by using more thantwo conjugate samples for estimation of a projection at a plane ofreconstruction (POR). This embodiment uses as many samples as thesampling pattern supports. The samples are located within apredetermined distance from the POR. Another feature of this embodimentis that non-integer pitch helical scans are performed for pitches thatare numerically less than the number of detector rows. In other words,if the number of detector rows is N, a pitch P:1 is used, where P is notan integer, and P<N. For example, a 2.5:1 pitch is used in oneembodiment having four detector rows so that no duplicated samples areacquired. Yet another feature of this embodiment is the use of nonlinearinterpolation techniques to preserve high frequency image components.Examples of suitable nonlinear interpolation include Lagrangeinterpolation and weighted interpolation-extrapolation, among others.Interpolation takes place by weighting projections prior to the filteredbackprojection.

In one embodiment and referring to FIG. 3, a system 10 having fourdetector rows R1, R2, R3, and R4 is used with a 2.5:1 helical pitch.Sampling patterns are depicted as a function of projection view angle,detector angle, and detector row. β1, β2, β3, and β4 represent viewangles at which corresponding detector rows R1, R2, R3, and R4 intersecta POR (not shown). These angles are arbitrarily selected, as only arelative angular span between them is important for any selected helicalpitch. For this embodiment, an angular distance between adjacent rowsthat intersect the POR is 0.8π. In FIG. 3, regions REG1, REG6, and REG11are examples of conjugate regions, because samples in these regionsdiffer in their view angles by either π or 2π. Each projection sample atthe plane of reconstruction is estimated based on conjugate samplesselected from three conjugate regions.

A “region 16” that would be located above REG15 could be included, sincethat region would also fall within a predetermined distance from POR.However, “region 16” is excluded in this embodiment to provide improvedtemporal resolution. For each set of conjugated regions, three regionsare included in this embodiment. If a “region 16” were included, theconjugate regions would include four regions 1, 6, 11 and 16. To providethe best temporal resolution, this embodiment seeks to formulate asmallest data set, in terms of view angle span, for imagereconstruction. Including “region 16” would increase the entire angularspan, as well as increase computation time and reduce uniformity.

Weights depend upon divisions of the region. To preserve symmetry andother properties, in one embodiment, projections are estimated along acurved plane written as β₁′=2.8π−γ, β₂′=2π−γ, β₃′=1.2π−γ, andβ₄′=0.4π−γ, where β₁′, β₂′, β₃′, and β₄′ represent view angles in acurved plane for corresponding detector rows R1, R2, R3, and R4,respectively. Each curved plane is defined by boundary conditions givenby regions cited in equations (1) to (4). An example of a curved planeis shown in FIG. 6. In this notation, γ represents the detector angle.Weights for third-order Lagrange interpolation for rows R1, R2, R3, andR4, respectively, are written as: $\begin{matrix}{{w_{1}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\frac{25}{6\pi^{2}}\left( {\beta - \beta_{1}^{\prime} + \frac{2\pi}{5}} \right)\left( {\beta - \beta_{1}^{\prime} + \frac{3\pi}{5}} \right)} & {{\beta_{1}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{1}^{\prime} - \frac{\pi}{5}}} \\{{- \frac{25}{2\pi^{2}}}\left( {\beta - \beta_{1}^{\prime} + \frac{2\pi}{5}} \right)\left( {\beta - \beta_{1}^{\prime} - \frac{\pi}{5}} \right)} & {{\beta_{1}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{1}^{\prime}} \\{\frac{25}{2\pi^{2}}\left( {\beta - \beta_{1}^{\prime} - \frac{2\pi}{5}} \right)\left( {\beta - \beta_{1}^{\prime} + \frac{\pi}{5}} \right)} & {\beta_{1}^{\prime} \leq \beta < {\beta_{1}^{\prime} + \frac{\pi}{5}}}\end{matrix} \right.} & (1)\end{matrix}$ $\begin{matrix}{{w_{2}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\frac{25}{3\pi^{2}}\left( {\beta - \beta_{2}^{\prime} + \frac{3\pi}{5}} \right)\left( {\beta - \beta_{2}^{\prime} + \frac{\pi}{5}} \right)} & {{\beta_{2}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{2}^{\prime} - \frac{\pi}{5}}} \\{{- \frac{25}{2\pi^{2}}}\left( {\beta - \beta_{2}^{\prime} - \frac{\pi}{5}} \right)\left( {\beta - \beta_{2}^{\prime} + \frac{\pi}{5}} \right)} & {{\beta_{2}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{2}^{\prime}} \\{\frac{25}{2\pi^{2}}\left( {\beta - \beta_{2}^{\prime} - \frac{2\pi}{5}} \right)\left( {\beta - \beta_{2}^{\prime} - \frac{\pi}{5}} \right)} & {\beta_{2}^{\prime} \leq \beta < {\beta_{2}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (2)\end{matrix}$ $\begin{matrix}{{w_{3}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\frac{25}{2\pi^{2}}\left( {\beta - \beta_{3}^{\prime} + \frac{\pi}{5}} \right)\left( {\beta - \beta_{3}^{\prime} + \frac{2\pi}{5}} \right)} & {{\beta_{3}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{3}^{\prime} - \frac{\pi}{5}}} \\{{- \frac{25}{\pi^{2}}}\left( {\beta - \beta_{3}^{\prime} - \frac{\pi}{5}} \right)\left( {\beta - \beta_{3}^{\prime} + \frac{\pi}{5}} \right)} & {{\beta_{3}^{\prime} - \frac{\pi}{5}} \leq \beta < {\beta_{3}^{\prime} + \frac{\pi}{5}}} \\{\frac{25}{3\pi^{2}}\left( {\beta - \beta_{3}^{\prime} - \frac{\pi}{5}} \right)\left( {\beta - \beta_{3}^{\prime} - \frac{3\pi}{5}} \right)} & {{\beta_{3}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{3}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (3)\end{matrix}$ $\begin{matrix}{{w_{4}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\frac{25}{2\pi^{2}}\left( {\beta - \beta_{4}^{\prime} + \frac{\pi}{5}} \right)\left( {\beta - \beta_{4}^{\prime} + \frac{2\pi}{5}} \right)} & {{\beta_{4}^{\prime} - \frac{2\pi}{5}} \leq \beta < \beta_{4}^{\prime}} \\{{- \frac{25}{2\pi^{2}}}\left( {\beta - \beta_{4}^{\prime} + \frac{\pi}{5}} \right)\left( {\beta - \beta_{4}^{\prime} - \frac{2\pi}{5}} \right)} & {\beta_{4}^{\prime} \leq \beta < {\beta_{4}^{\prime} + \frac{\pi}{5}}} \\{\frac{25}{6\pi^{2}}\left( {\beta - \beta_{4}^{\prime} - \frac{3\pi}{5}} \right)\left( {\beta - \beta_{4}^{\prime} - \frac{2\pi}{5}} \right)} & {{\beta_{4}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{4}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (4)\end{matrix}$

In one embodiment of the present invention, weightedinterpolation-extrapolation (WIE), i.e., a combining of weightedinterpolation measurements with weighted extrapolated measurements, isused prior to the filtered backprojection. FIG. 4 illustrates an examplehaving three sampling points that are located on both sides of a point xwhere the interpolation is to take place. (A different case in which onesample is located at left and two samples at the right is treatedsimilarly.) For one known helical reconstruction algorithm, only x2 andx3 are used for linear interpolation represented by line 50. Weights t2and t3 used for x2 and x3, respectively, are written as: $\begin{matrix}{{{t2} = \frac{{x3} - x}{{x3} - {x2}}}{{t3} = \frac{x - {x2}}{{x3} - {x2}}}} & (5)\end{matrix}$

In one embodiment of the present invention, to use points x1 and x2 toestimate x, extrapolation is used as represented by line 52. Weights forx1 and x2 (denoted by e1 and e2) are written as: $\begin{matrix}{{{e1} = \frac{{x2} - x}{{x2} - {x1}}}{{e2} = \frac{x - {x1}}{{x2} - {x1}}}} & (6)\end{matrix}$

In this notation, x1, x2, and x3 are x-axis coordinates of locations ofmeasured signals, and x is the location of the signal to be estimated.The weights used in linear interpolation are calculated based on arelative distance of the two points to the interpolation location.Combined weights for points x1, x2, and x3 (denoted by q1, q2, and q3,respectively) are written as:q1=(1−t ₂ ^(α) −t ₃ ^(α))e1,q2=(1−t ₂ ^(α) −t ₃ ^(α))e2+(t ₂ ^(α) +t ₃ ^(α))t2,q3=(t ₂ ^(α) +t ₃ ^(α))t3.   (7)

In equation (7), α is a parameter that adjusts relative strength andcontributions of the extrapolation. The terms q1, q2, and q3 representweights for the three measured samples to estimate a sample in the POR.

This interpolation scheme overcomes the shortcomings of linearinterpolation and enables a better preservation of high frequencyinformation contents. Weighting functions for rows R1, R2, R3, and R4,respectively, for this interpolation scheme, represented by line 54, arewritten as: $\begin{matrix}{{w_{1}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},} & {{\beta_{1}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{1}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},}\end{matrix} & {{\beta_{1}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{1}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{1}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{1}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)},} & {\beta_{1}^{\prime} \leq \beta < {\beta_{1}^{\prime} + \frac{\pi}{5}}}\end{matrix} \right.} & (8)\end{matrix}$where θ₁=β−β₁′=β−2.8π+γ, $\begin{matrix}{{w_{2}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + {3\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{{- 5}\theta_{2}} - \pi}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},} & {{\beta_{2}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{2}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{2}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{2}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},}\end{matrix} & {{\beta_{2}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{2}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{2}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {\beta_{2}^{\prime} \leq \beta < {\beta_{2}^{\prime} + \frac{\pi}{5}}} \\{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} - \pi}{\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {{\beta_{2}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{2}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (9)\end{matrix}$where θ₂=β−β₂′=β−2π+γ, $\begin{matrix}{{w_{3}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{3}} + {2\pi}}{\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{{- 5}\theta_{3}} - \pi}{\pi} \right)^{\alpha} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{3}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{3}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} - \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)}\end{matrix} & {\beta_{3}^{\prime} \leq \beta < {\beta_{3}^{\prime} + \frac{\pi}{5}}} \\\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{3}} - \pi}{2\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{3\pi} - {5\theta_{3}}}{2\pi} \right)^{\alpha} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{3}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (10)\end{matrix}$where θ₃=β−β₃′=β−1.2π+γ, and $\begin{matrix}{{w_{4}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{4}} + {2\pi}}{\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{{- 5}\theta_{4}} - \pi}{\pi} \right)^{\alpha} \right\rbrack\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{4}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{4}^{\prime} - \frac{\pi}{5}}} \\{\left\lbrack {\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{4}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} & {{\beta_{4}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{4}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} + \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)}\end{matrix} & {\beta_{4}^{\prime} \leq \beta < {\beta_{4}^{\prime} + \frac{\pi}{5}}} \\\begin{matrix}\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} +} \right. \\{\left. \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)}\end{matrix} & {{\beta_{4}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{4}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.} & (11)\end{matrix}$where θ₄=β−β₄′=β−0.4π+γ,

In equations (8)–(11), w1, w2, w3, and w4 are derived weights for fourrows, based on conjugate regions shown in FIG. 3 and in equation (7).For each sample to be estimated on POR, three conjugate samples areselected, based on FIG. 3, and equation (7) is applied to determinetheir weights. The three conjugate samples could, for example, come fromany three of R1, R2, R3, and R4.

To demonstrate the advantages of the use of the above reconstructionweights expressed in equations (8)–(11), a phantom study was performed.In this study, a thin plate placed parallel to an x-y plane inside a 20cm poly phantom was scanned with 120 kV/200 mA/1.25 mm/1.25 sec at both3:1 and 2.5:1 pitches. Images were reconstructed every 0.1 mm to ensureadequate samples to map out slice profiles. For the 3:1 helical pitch, aknown reconstruction algorithm was used. For the 2.5:1 helical pitch,the weights expressed in equations (8)–(11) were used. Profiles of thethin plates (and therefore, a system slice sensitivity profile) werecalculated for both cases. To avoid effects of statistical noise due tofinite x-ray photon statistics, each point on a profile curve is anaverage over a 21 by 21 pixel region (in an x-y plane) centered on thethin plate. FIG. 5 depicts a slice profile A for a 3:1 helical pitchimage reconstructed using the known reconstruction algorithm and a sliceprofile B for a 2.5:1 helical pitch image reconstructed using the weightfunctions expressed in equations (8)–(11). A much narrower slice profileis obtained and the peak intensity for the profile is higher whenequations (8)–(11) are used.

To more fully evaluate a WIE reconstruction embodiment of the presentinvention, a noise analysis was performed. For comparison, images in zof WIE were smoothed by a filter kernel such that WIE with smoothingprovides the same slice profile as the known algorithm withoutsmoothing. A kernel that satisfies this condition has been found to bean 11-point kernel with coefficients [0.5, 1, 1, . . . , 1, 1, 0.5].

Image smoothing (in z) for the WIE images was also performed for a casein which the phantom was uniform in z. The standard deviation of thesmoothed image was determined to be 5.23 HU. The standard deviationobserved using the known algorithm was 6.63 HU. This result indicatesthat for the same slice sensitivity profile, a nearly 37% mA reductioncan be achieved, if the noise is to be maintained at the same level. Inimages corresponding to the location where the thin plate is located,WIE-weighted images with smoothing show less image artifacts than imagesreconstructed using the known reconstruction algorithm.

In another embodiment, an equivalent result to a non-linearinterpolation is achieved by multiplying a set of projection date by aset of weights. The weights, for a third order Lagrange interpolation,are written as equations (1)–(4), while weights for WEE are written asequations (8)–(11).

It will thus be seen that embodiments of the present invention providebetter sampling patterns, acquire less redundant information, result inless suppression of sharp structures of object and better slicesensitivity profiles than known helical CT image reconstruction methodsand systems.

Although the exemplary embodiments described herein and having testresults presented herein are 4-slice embodiments, other embodiments ofthe invention are applicable to multi-slice CT imaging systems providingdifferent numbers of image slices. For example, in other embodiments,8-slice, 16-slice, 32-slice, etc. CT imaging systems are used. Inaddition, the invention is not limited to embodiments providing a 2.5:1helical pitch. This pitch was selected for purposes of illustrationonly. Other embodiments of the invention are applicable to CT imagingsystems providing different helical pitches. In addition, the CT systemdescribed herein is a “third generation” system in which both the x-raysource and detector rotate with the gantry. Many other CT systemsincluding “fourth generation” systems wherein the detector is afull-ring stationary detector and only the x-ray source rotates with thegantry, may be used if individual detector elements are corrected toprovide substantially uniform responses to a given x-ray beam.

While the invention has been described in terms of various specificembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theclaims. Accordingly, the spirit and scope of the invention are to belimited only by the terms of the appended claims.

1. A method for imaging an object with a computed tomographic (CT)imaging system, comprising the steps of: helically scanning the objectwith a multi-slice CT imaging system having a plurality of detector rowsto acquire attenuation measurements of the object, the measurementsincluding more than two conjugate samples, wherein a difference betweena view angle of one of the more than two conjugate samples and a viewangle of any one of the remaining conjugate samples of the more than twoconjugate samples is n π, wherein n is an integer greater than zero;estimating at least one projection along a curved plane ofreconstruction of the object using the attenuation measurements of theobject, including the more than two conjugate samples; applyingdifferent weighting functions within each detector row dependent upon adivision of conjugate regions for each detector row and dependent uponview angles in a curved plane for corresponding detector rows; andfiltering and backprojecting the attenuation measurements of the object,including the more than two conjugate samples, to reconstruct at leastone image slice of the object.
 2. A method in accordance with claim 1wherein the more than two conjugate samples are located within apredetermined distance from the curved plane of reconstruction of theobject.
 3. A method in accordance with claim 1 wherein the CT imagingsystem has N detector rows, and further comprising the step of selectinga helical pitch P:1 for said helical scan, where P is a non-integer lessthan N.
 4. A method in accordance with claim 3 wherein N=4 and P=2.5. 5.A method in accordance with claim 1 further comprising the step ofapplying a non-linear interpolation to the attenuation measurementsprior to said filtering and backprojecting.
 6. A method in accordancewith claim 5 wherein applying a non-linear interpolation to theattenuation measurements comprises applying a Lagrange interpolation tothe attenuation measurements.
 7. A method in accordance with claim 6wherein applying a Lagrange interpolation to the attenuationmeasurements comprises applying third order Lagrange interpolationweights to measurements from four detector rows.
 8. A method inaccordance with claim 5 wherein the CT imaging system has 4 detectorrows, helically scanning the object to obtain attenuation measurementscomprises the step of helically scanning the object at a pitch of 2.5:1,and said estimating at least one projection along the curved plane ofreconstruction comprises the step of estimating projections along thecurved plane of reconstruction written as β₁′=2.8π−γ, β₂′=2π−γ,β₃′=1.2π−γ, and β₄′=0.4π−γ, where β₁′, β₂′, β₃′, and β₄′ represent viewangles in a curved plane for corresponding detector rows R1, R2, R3, andR4, respectively, and γ represents a detector angle.
 9. A method inaccordance with claim 5 wherein applying a non-linear interpolation tothe attenuation measurements comprises combining weighted interpolatedmeasurements with weighted extrapolated measurements.
 10. A method inaccordance with claim 9 wherein the CT imaging system has 4 detectorrows R1, R2, R3, and R4, helically scanning the object to obtainattenuation measurements comprises the step of helically scanning theobject at a pitch of 2.5:1, and said method further comprises the stepof applying weights to attenuation measurements for detector rows R1,R2, R3, and R4 respectively, wherein the applied weights are written as:${w_{1}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},} & {{\beta_{1}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{1}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},}\end{matrix} & {{\beta_{1}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{1}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{1}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{1}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)},} & {\beta_{1}^{\prime} \leq \beta < {\beta_{1}^{\prime} + \frac{\pi}{5}}}\end{matrix} \right.$ where θ₁=β−β₁′=β−2.8π+γ,${w_{2}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + {3\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{{- 5}\theta_{2}} - \pi}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},} & {{\beta_{2}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{2}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{2}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{2}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},}\end{matrix} & {{\beta_{2}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{2}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{2}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {\beta_{2}^{\prime} \leq \beta < {\beta_{2}^{\prime} + \frac{\pi}{5}}} \\{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} - \pi}{\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {{\beta_{2}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{2}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₂=β−β₂′=β−2π+γ,${w_{3}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{3}} + {2\pi}}{\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{{- 5}\theta_{3}} - \pi}{\pi} \right)^{\alpha} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{3}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{3}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} - \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)}\end{matrix} & {\beta_{3}^{\prime} \leq \beta < {\beta_{3}^{\prime} + \frac{\pi}{5}}} \\\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{3}} - \pi}{2\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{3\pi} - {5\theta_{3}}}{2\pi} \right)^{\alpha} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{3}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₃=β−β₃′=β−1.2π+γ, and${w_{4}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}\begin{matrix}\left\lbrack {1 - \left( \frac{{5\theta_{4}} + {2\pi}}{\pi} \right)^{\alpha} -} \right. \\{\left. \left( \frac{{{- 5}\theta_{4}} - \pi}{\pi} \right)^{\alpha} \right\rbrack\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{4}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{4}^{\prime} - \frac{\pi}{5}}} \\{\left\lbrack {\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{4}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} & {{\beta_{4}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{4}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} + \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)}\end{matrix} & {\beta_{4}^{\prime} \leq \beta < {\beta_{4}^{\prime} + \frac{\pi}{5}}} \\\begin{matrix}\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} +} \right. \\{\left. \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)}\end{matrix} & {{\beta_{4}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{4}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₄=β−β₄′=β−0.4π+γ, β₁′=2.8π−γ, β₂′=2π−γ,β₃′=1.2π−γ, and β₄′=0.4π−γ; β₁′β₂′β₃′, and β₄′ represent view anglesintersecting the POR for detector rows R1, R2, R3, and R4, respectively,and γ represents a detector angle.
 11. A method in accordance with claim1 further comprising the step of applying a set of weights to theattenuation measurements prior to said filtering and backprojecting. 12.A method in accordance with claim 11 wherein applying a set of weightsto the attenuation measurements comprises the step of applying Lagrangeweights to the attenuation measurements.
 13. A method in accordance withclaim 12 wherein applying Lagrange weights to the attenuationmeasurements comprises applying third order Lagrange weights tomeasurements from four detector rows.
 14. A method in accordance withclaim 11 wherein the CT imaging system has 4 detector rows, helicallyscanning the object to obtain attenuation measurements comprises thestep of helically scanning the object at a pitch of 2.5:1, and saidestimating at least one projection along the curved plane ofreconstruction comprises the step of estimating projections along thecurved plane of reconstruction written as β₁′=2.8π−γ, β₂′=2π−γ,β₃′=1.2π−γ, and β₄′=0.4π−γ, where β₁′, β₂′, β₃′, and β₄′ represent viewangles in a curved plane for corresponding detector rows R1, R2, R3, andR4, respectively, and γ represents a detector angle.
 15. A method inaccordance with claim 1 further comprising the step of applyinginterpolation and extrapolation to determine weights to be applied tothe attenuation measurements.
 16. A computed tomographic (CT) imagingsystem for imaging an object, said system comprising a radiation sourceand a multi-slice detector having a plurality of detector rows, saidmulti-slice detector configured to acquire attenuation measurements ofan object between said radiation source and said multi-slice detector,said system configured to: helical scan the object to acquireattenuation measurements of the object, said measurements including morethan two conjugate samples, wherein a difference between a view angle ofone of the more than two conjugate samples and a view angle of any oneof the remaining conjugate samples of the more than two conjugatesamples is n π, wherein n is an integer greater than zero; estimate atleast one projection along a curved plane of reconstruction of theobject using the attenuation measurements of the object, including themore than two conjugate samples; applying different weighting functionswithin each detector row dependent upon a division of conjugate regionsfor each detector row and dependent upon view angles in a curved planefor corresponding detector rows: and filter and backproject theattenuation measurements of the object, including the more than twoconjugate samples, to reconstruct at least one image slice of theobject.
 17. A system in accordance with claim 16 further configured sothat the more than two conjugate samples are located within apredetermined distance from the curved plane of reconstruction of theobject.
 18. A system in accordance with claim 17 having N detector rows,and further configured to perform the helical scan at a pitch P:1, whereP is a non-integer less than N.
 19. A system in accordance with claim 18wherein N=4 and P=2.5.
 20. A system in accordance with claim 16 furtherconfigured to apply a non-linear interpolation to the attenuationmeasurements prior to said filtering and backprojecting.
 21. A system inaccordance with claim 20 wherein said system being configured to apply anon-linear interpolation to the attenuation measurements comprises saidsystem being configured to apply a Lagrange interpolation to theattenuation measurements.
 22. A system in accordance with claim 21wherein said system being configured to apply a Lagrange interpolationto the attenuation measurements comprises said system being configuredto apply third order Lagrange interpolation weights to measurements fromfour detector rows.
 23. A system in accordance with claim 20 having 4detector rows, and wherein said system being configured to helicallyscan the object to obtain attenuation measurements comprises said systembeing configured to helically scan the object at a pitch of 2.5:1, andto estimate at least one projection along the curved plane ofreconstruction said system is further configured to estimate projectionsalong the curved plane of reconstruction written as β₁′=2.8π−γ,β₂′=2π−γ, β₃′=1.2π−γ, and β₄′=0.4π−γ, where β₁′, β₂′, β₃′, and β₄′represent view angles in a curved plane for corresponding detector rowsR1, R2, R3, and R4, respectively, and γ represents a detector angle. 24.A system in accordance with claim 20 wherein said system beingconfigured to apply a non-linear interpolation to the attenuationmeasurements comprises said system being configured to combine weightedinterpolated measurements with weighted extrapolated measurements.
 25. Asystem in accordance with claim 24 having 4 detector rows R1, R2, R3,and R4, wherein said system being configured to helically scan theobject to obtain attenuation measurements comprises said system beingconfigured to helically scan the object at a pitch of 2.5:1, and saidsystem is further configured to apply weights to attenuationmeasurements for detector rows R1, R2, R3, and R4 respectively, whereinthe applied weights are written as:${w_{1}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},} & {{\beta_{1}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{1}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{1}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{1}} + {2\pi}}{2\pi} \right)},}\end{matrix} & {{\beta_{1}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{1}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{1}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{1}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{1}}}{\pi} \right)},} & {\beta_{1}^{\prime} \leq \beta < {\beta_{1}^{\prime} + \frac{\pi}{5}}}\end{matrix} \right.$ where θ₁=β−β₁′=β−2.8π+γ,${w_{2}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + {3\pi}}{2\pi} \right)^{\alpha} - \left( \frac{{{- 5}\theta_{2}} - \pi}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},} & {{\beta_{2}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{2}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{2}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)} +} \\{{\left\lbrack {\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{2}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{2}} + \pi}{\pi} \right)},}\end{matrix} & {{\beta_{2}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{2}^{\prime}} \\{{\left\lbrack {\left( \frac{5\theta_{2}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {\beta_{2}^{\prime} \leq \beta < {\beta_{2}^{\prime} + \frac{\pi}{5}}} \\{{\left\lbrack {1 - \left( \frac{{5\theta_{2}} - \pi}{\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{2}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{2}}}{\pi} \right)},} & {{\beta_{2}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{2}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₂=β−β₂′=β−2π+γ,${w_{3}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\left\lbrack {1 - \left( \frac{{5\theta_{3}} + {2\pi}}{\pi} \right)^{\alpha} - \left( \frac{{{- 5}\theta_{3}} - \pi}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)} & {{\beta_{3}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{3}^{\prime} - \frac{\pi}{5}}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} - \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{3}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{3}} + \pi}{\pi} \right)}\end{matrix} & {{\beta_{3}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{3}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} - \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{3}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{3}}{\pi} \right)^{\alpha} + \left( \frac{\pi - {5\theta_{3}}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)}\end{matrix} & {\beta_{3}^{\prime} \leq \beta < {\beta_{3}^{\prime} + \frac{\pi}{5}}} \\{\left\lbrack {1 - \left( \frac{{5\theta_{3}} - \pi}{2\pi} \right)^{\alpha} - \left( \frac{{3\pi} - {5\theta_{3}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi - {5\theta_{3}}}{\pi} \right)} & {{\beta_{3}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{3}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₃=β−β₃′=β−1.2π+γ, and${w_{4}\left( {\gamma,\beta} \right)} = \left\{ \begin{matrix}{\left\lbrack {1 - \left( \frac{{5\theta_{4}} + {2\pi}}{\pi} \right)^{\alpha} - \left( \frac{{{- 5}\theta_{4}} - \pi}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)} & {{\beta_{4}^{\prime} - \frac{2\pi}{5}} \leq \beta < {\beta_{4}^{\prime} - \frac{\pi}{5}}} \\{\left\lbrack {\left( \frac{{5\theta_{4}} + \pi}{\pi} \right)^{\alpha} + \left( \frac{{- 5}\theta_{4}}{\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} & {{\beta_{4}^{\prime} - \frac{\pi}{5}} \leq \beta < \beta_{4}^{\prime}} \\\begin{matrix}{{\left\lbrack {1 - \left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} - \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{\pi + {5\theta_{4}}}{\pi} \right)} +} \\{\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} + \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)}\end{matrix} & {\beta_{4}^{\prime} \leq \beta < {\beta_{4}^{\prime} + \frac{\pi}{5}}} \\{\left\lbrack {\left( \frac{5\theta_{4}}{2\pi} \right)^{\alpha} + \left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)^{\alpha}} \right\rbrack\left( \frac{{2\pi} - {5\theta_{4}}}{2\pi} \right)} & {{\beta_{4}^{\prime} + \frac{\pi}{5}} \leq \beta < {\beta_{4}^{\prime} + \frac{2\pi}{5}}}\end{matrix} \right.$ where θ₄=β−β₄′=β−0.4π+γ, β₁′=2.8π−γ, β₂′=2π−γ,β₃′=1.2π−γ, and β₄′=0.4π−γ; β₁′, β₂′, β₃′, and β₄′ represent view anglesintersecting the POR for detector rows R1, R2, R3, and R4, respectively,and γ represents a detector angle.
 26. A system in accordance with claim16 further configured to apply a set of weights to the attenuationmeasurements prior to said filtering and backprojecting.
 27. A system inaccordance with claim 26 wherein said system being configured to apply aset of weights to the attenuation measurements comprises said systembeing configured to apply Lagrange weights to the attenuationmeasurements.
 28. A system in accordance with claim 27 wherein saidsystem being configured to apply Lagrange weights to the attenuationmeasurements comprises said system being configured to apply third orderLagrange weights to measurements from four detector rows.
 29. A systemin accordance with claim 16 having 4 detector rows, and said systembeing configured to helically scan the object to obtain attenuationmeasurements comprises said system being configured to helically scanthe object at a pitch of 2.5:1, and to estimate at least one projectionalong the curved plane of reconstruction said system is furtherconfigured to estimate projections along the curved plane ofreconstruction written as β₁′=2.8π−γ, β₂′=2π−γ, β₃′=1.2π−γ, andβ₄′=0.4π−γ, where β₁′, β₂′, β₃′, and β₄′ represent view angles in acurved plane for corresponding detector rows R1, R2, R3, and R4,respectively, and γ represents a detector angle.
 30. A system inaccordance with claim 16 further configured to apply interpolation andextrapolation to determine weights to be applied to the attenuationmeasurements.